To solve the problem, we'll follow a step-by-step approach, using the given condition \( x + \frac{1}{x} = \sqrt{5} \). We need to find values for \( x^2 + \frac{1}{x^2} \) and \( x^4 + \frac{1}{x^4} \).
### Step 1: Finding \( x^2 + \frac{1}{x^2} \)
Let's start by squaring both sides of the given equation:
[tex]\[ \left( x + \frac{1}{x} \right)^2 = (\sqrt{5})^2 \][/tex]
Expanding the left-hand side:
[tex]\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 5 \][/tex]
Since \( x \cdot \frac{1}{x} = 1 \), we simplify the equation to:
[tex]\[ x^2 + 2 + \frac{1}{x^2} = 5 \][/tex]
Subtract 2 from both sides:
[tex]\[ x^2 + \frac{1}{x^2} = 3 \][/tex]
### Step 2: Finding \( x^4 + \frac{1}{x^4} \)
To find \( x^4 + \frac{1}{x^4} \), we square \( x^2 + \frac{1}{x^2} \):
[tex]\[ \left( x^2 + \frac{1}{x^2} \right)^2 \][/tex]
We already found that \( x^2 + \frac{1}{x^2} = 3 \). Squaring this:
[tex]\[ (3)^2 = x^4 + 2 \cdot x^2 \cdot \frac{1}{x^2} + \frac{1}{x^4} \][/tex]
Since \( x^2 \cdot \frac{1}{x^2} = 1 \), we simplify the equation:
[tex]\[ 9 = x^4 + 2 + \frac{1}{x^4} \][/tex]
Subtract 2 from both sides:
[tex]\[ x^4 + \frac{1}{x^4} = 7 \][/tex]
### Conclusion
Based on the steps we followed:
[tex]\[ x^2 + \frac{1}{x^2} = 3 \][/tex]
[tex]\[ x^4 + \frac{1}{x^4} = 7 \][/tex]
Thus, we have found the required values as follows:
- \( x^2 + \frac{1}{x^2} = 3 \)
- [tex]\( x^4 + \frac{1}{x^4} = 7 \)[/tex]
